New compact representations of infinite graphs are
investigated. Finite automata are used to represent labelled hyper-graphs
which can be also multi-graphs. Our approach consists of a general
framework where vertices are represented by a regular prefix-free language and
edges are represented by a regular language and a function over tuples.
We consider three different functions over tuples:
given a tuple
the first function returns its
first difference, the second one returns its suffix and
the last one returns its infixes.
The first-difference function is substantially
a direct generalization to infinite multi-hyper-graphs of the
representation introduced by Ehrenfeucht et al. for finite graphs.
This representation, though very interesting for finite graphs, turns out
to be quite unsatisfactory for infinite graphs.
The other two functions we consider while preserving some interesting
features of their representation also achieves a high expressive power.
As a matter of fact,
our formalism either with the suffix or infix function results
to be more powerful than the
equational graphs introduced by Courcelle and the simple graphs
defined by Caucal.
The monadic second order theories
of these two classes of graphs are undecidable,
but still many interesting graph properties are
decidable. The use of a regular
prefix-free language to represent the vertices allows (fixed the language of
the edges) to express a graph by a labelled tree,
moreover, the use of finite automata to represent the edges allows
the verification of graph properties.

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